Learning hydrodynamic equations for active matter from particle simulations and experiments.

Recent advances in high-resolution imaging techniques and particle-based simulation methods have enabled the precise microscopic characterization of collective dynamics in various biological and engineered active matter systems. In parallel, data-driven algorithms for learning interpretable continuum models have shown promising potential for the recovery of underlying partial differential equations (PDEs) from continuum simulation data. By contrast, learning macroscopic hydrodynamic equations for active matter directly from experiments or particle simulations remains a major challenge, especially when continuum models are not known a priori or analytic coarse graining fails, as often is the case for nondilute and heterogeneous systems. Here, we present a framework that leverages spectral basis representations and sparse regression algorithms to discover PDE models from microscopic simulation and experimental data, while incorporating the relevant physical symmetries. We illustrate the practical potential through a range of applications, from a chiral active particle model mimicking nonidentical swimming cells to recent microroller experiments and schooling fish. In all these cases, our scheme learns hydrodynamic equations that reproduce the self-organized collective dynamics observed in the simulations and experiments. This inference framework makes it possible to measure a large number of hydrodynamic parameters in parallel and directly from video data.

An additive algorithm for origami design.

Inspired by the allure of additive fabrication, we pose the problem of origami design from a different perspective: How can we grow a folded surface in three dimensions from a seed so that it is guaranteed to be isometric to the plane? We solve this problem in two steps: by first identifying the geometric conditions for the compatible completion of two separate folds into a single developable fourfold vertex, and then showing how this foundation allows us to grow a geometrically compatible front at the boundary of a given folded seed. This yields a complete marching, or additive, algorithm for the inverse design of the complete space of developable quad origami patterns that can be folded from flat sheets. We illustrate the flexibility of our approach by growing ordered, disordered, straight, and curved-folded origami and fitting surfaces of given curvature with folded approximants. Overall, our simple shift in perspective from a global search to a local rule has the potential to transform origami-based metastructure design.

Geometry and dynamics link form, function, and evolution of finch beaks.

Darwin's finches are a classic example of adaptive radiation, exemplified by their adaptive and functional beak morphologies. To quantify their form, we carry out a morphometric analysis of the three-dimensional beak shapes of all of Darwin's finches and find that they can be fit by a transverse parabolic shape with a curvature that increases linearly from the base toward the tip of the beak. The morphological variation of beak orientation, aspect ratios, and curvatures allows us to quantify beak function in terms of the elementary theory of machines, consistent with the dietary variations across finches. Finally, to explain the origin of the evolutionary morphometry and the developmental morphogenesis of the finch beak, we propose an experimentally motivated growth law at the cellular level that simplifies to a variant of curvature-driven flow at the tissue level and captures the range of observed beak shapes in terms of a simple morphospace. Altogether, our study illuminates how a minimal combination of geometry and dynamics allows for functional form to develop and evolve.

Beak morphometry and morphogenesis across avian radiations.

Adaptive avian radiations associated with the diversification of bird beaks into a multitude of forms enabling different functions are exemplified by Darwin's finches and Hawaiian honeycreepers. To elucidate the nature of these radiations, we quantified beak shape and skull shape using a variety of geometric measures that allowed us to collapse the variability of beak shape into a minimal set of geometric parameters. Furthermore, we find that just two measures of beak shape-the ratio of the width to length and the normalized sharpening rate (increase in the transverse beak curvature near the tip relative to that at the base of the beak)-are strongly correlated with diet. Finally, by considering how transverse sections to the beak centreline evolve with distance from the tip, we show that a simple geometry-driven growth law termed 'modified mean curvature flow' captures the beak shapes of Darwin's finches and Hawaiian honeycreepers. A surprising consequence of the simple growth law is that beak shapes that are not allowed based on the developmental programme of the beak are also not observed in nature, suggesting a link between evolutionary morphology and development in terms of growth-driven developmental constraints.

Combined measures of mimetic fidelity explain imperfect mimicry in a brood parasite-host system.

The persistence of imperfect mimicry in nature presents a challenge to mimicry theory. Some hypotheses for the existence of imperfect mimicry make differing predictions depending on how mimetic fidelity is measured. Here, we measure mimetic fidelity in a brood parasite-host system using both trait-based and response-based measures of mimetic fidelity. Cuckoo finches Anomalospiza imberbis lay imperfectly mimetic eggs that lack the fine scribbling characteristic of eggs of the tawny-flanked prinia Prinia subflava, a common host species. A trait-based discriminant analysis based on Minkowski functionals-that use geometric and topological morphometric methods related to egg pattern shape and coverage-reflects this consistent difference between host and parasite eggs. These methods could be applied to quantify other phenotypes including stripes and waved patterns. Furthermore, by painting scribbles onto cuckoo finch eggs and testing their rate of rejection compared to control eggs (i.e. a response-based approach to quantify mimetic fidelity), we show that prinias do not discriminate between eggs based on the absence of scribbles. Overall, our results support relaxed selection on cuckoo finches to mimic scribbles, since prinias do not respond differently to eggs with and without scribbles, despite the existence of this consistent trait difference.

Deterministic and stochastic control of kirigami topology.

Kirigami, the creative art of paper cutting, is a promising paradigm for mechanical metamaterials. However, to make kirigami-inspired structures a reality requires controlling the topology of kirigami to achieve connectivity and rigidity. We address this question by deriving the maximum number of cuts (minimum number of links) that still allow us to preserve global rigidity and connectivity of the kirigami. A deterministic hierarchical construction method yields an efficient topological way to control both the number of connected pieces and the total degrees of freedom. A statistical approach to the control of rigidity and connectivity in kirigami with random cuts complements the deterministic pathway, and shows that both the number of connected pieces and the degrees of freedom show percolation transitions as a function of the density of cuts (links). Together, this provides a general framework for the control of rigidity and connectivity in planar kirigami.

Programming shape using kirigami tessellations.

Kirigami tessellations, regular planar patterns formed by partially cutting flat, thin sheets, allow compact shapes to morph into open structures with rich geometries and unusual material properties. However, geometric and topological constraints make the design of such structures challenging. Here we pose and solve the inverse problem of determining the number, size and orientation of cuts that enables the deployment of a closed, compact regular kirigami tessellation to conform approximately to any prescribed target shape in two or three dimensions. We first identify the constraints on the lengths and angles of generalized kirigami tessellations that guarantee that their reconfigured face geometries can be contracted from a non-trivial deployed shape to a compact, non-overlapping planar cut pattern. We then encode these conditions into a flexible constrained optimization framework to obtain generalized kirigami patterns derived from various periodic tesselations of the plane that can be deployed into a wide variety of prescribed shapes. A simple mechanical analysis of the resulting structure allows us to determine and control the stability of the deployed state and control the deployment path. Finally, we fabricate physical models that deploy in two and three dimensions to validate this inverse design approach. Altogether, our approach, combining geometry, topology and optimization, highlights the potential for generalized kirigami tessellations as building blocks for shape-morphing mechanical metamaterials.

Self-Excited Motions of Volatile Drops on Swellable Sheets.

When a volatile droplet is deposited on a floating swellable sheet, it becomes asymmetric, lobed and mobile. We describe and quantify this phenomena that involves nonequilibrium swelling, evaporation and motion, working together to realize a self-excitable spatially extended oscillator. Solvent penetration causes the film to swell locally and eventually buckle, changing its shape and the drop responds by moving. Simultaneously, solvent evaporation from the swollen film causes it to regain its shape once the droplet has moved away. The process repeats and leads to complex pulsatile spinning and/or sliding movements. We use a one-dimensional experiment to highlight the slow swelling of and evaporation from the film and the fast motion of the drop, a characteristic of excitable systems. Finally, we provide a phase diagram for droplet excitability as a function of drop size and film thickness and scaling laws for the motion of the droplet.

In Vivo Microrheology Reveals Local Elastic and Plastic Responses Inside 3D Bacterial Biofilms.

Bacterial biofilms are highly abundant 3D living materials capable of performing complex biomechanical and biochemical functions, including programmable growth, self-repair, filtration, and bioproduction. Methods to measure internal mechanical properties of biofilms in vivo with spatial resolution on the cellular scale have been lacking. Here, thousands of cells are tracked inside living 3D biofilms of the bacterium Vibrio cholerae during and after the application of shear stress, for a wide range of stress amplitudes, periods, and biofilm sizes, which revealed anisotropic elastic and plastic responses of both cell displacements and cell reorientations. Using cellular tracking to infer parameters of a general mechanical model, spatially-resolved measurements of the elastic modulus inside the biofilm are obtained, which correlate with the spatial distribution of the polysaccharides within the biofilm matrix. The noninvasive microrheology and force-inference approach introduced here provides a general framework for studying mechanical properties with high spatial resolution in living materials.

Automatic landmark detection and registration of brain cortical surfaces via quasi-conformal geometry and convolutional neural networks.

In medical imaging, surface registration is extensively used for performing systematic comparisons between anatomical structures, with a prime example being the highly convoluted brain cortical surfaces. To obtain a meaningful registration, a common approach is to identify prominent features on the surfaces and establish a low-distortion mapping between them with the feature correspondence encoded as landmark constraints. Prior registration works have primarily focused on using manually labeled landmarks and solving highly nonlinear optimization problems, which are time-consuming and hence hinder practical applications. In this work, we propose a novel framework for the automatic landmark detection and registration of brain cortical surfaces using quasi-conformal geometry and convolutional neural networks. We first develop a landmark detection network (LD-Net) that allows for the automatic extraction of landmark curves given two prescribed starting and ending points based on the surface geometry. We then utilize the detected landmarks and quasi-conformal theory for achieving the surface registration. Specifically, we develop a coefficient prediction network (CP-Net) for predicting the Beltrami coefficients associated with the desired landmark-based registration and a mapping network called the disk Beltrami solver network (DBS-Net) for generating quasi-conformal mappings from the predicted Beltrami coefficients, with the bijectivity guaranteed by quasi-conformal theory. Experimental results are presented to demonstrate the effectiveness of our proposed framework. Altogether, our work paves a new way for surface-based morphometry and medical shape analysis.

An additive framework for kirigami design.

We present an additive approach for the inverse design of kirigami-based mechanical metamaterials by focusing on the empty (negative) spaces instead of the solid tiles. By considering each negative space as a four-bar linkage, we identify a simple recursive relationship between adjacent linkages, yielding an efficient method for creating kirigami patterns. This allows us to solve the kirigami design problem using elementary linear algebra, with compatibility, reconfigurability and rigid-deployability encoded into an iterative procedure involving simple matrix multiplications. The resulting linear design strategy circumvents the solution of a non-convex global optimization problem and allows us to control the degrees of freedom in the deployment angle field, linkage offsets and boundary conditions. We demonstrate this by creating a large variety of rigid-deployable, compact, reconfigurable kirigami patterns. We then realize our kirigami designs physically using two simple but effective fabrication strategies with very different materials. Altogether, our additive approaches present routes for efficient mechanical metamaterial design and fabrication based on ori/kirigami art forms.

Adaptive area-preserving parameterization of open and closed anatomical surfaces.

The parameterization of open and closed anatomical surfaces is of fundamental importance in many biomedical applications. Spherical harmonics, a set of basis functions defined on the unit sphere, are widely used for anatomical shape description. However, establishing a one-to-one correspondence between the object surface and the entire unit sphere may induce a large geometric distortion in case the shape of the surface is too different from a perfect sphere. In this work, we propose adaptive area-preserving parameterization methods for simply-connected open and closed surfaces with the target of the parameterization being a spherical cap. Our methods optimize the shape of the parameter domain along with the mapping from the object surface to the parameter domain. The object surface will be globally mapped to an optimal spherical cap region of the unit sphere in an area-preserving manner while also exhibiting low conformal distortion. We further develop a set of spherical harmonics-like basis functions defined over the adaptive spherical cap domain, which we call the adaptive harmonics. Experimental results show that the proposed parameterization methods outperform the existing methods for both open and closed anatomical surfaces in terms of area and angle distortion. Surface description of the object surfaces can be effectively achieved using a novel combination of the adaptive parameterization and the adaptive harmonics. Our work provides a novel way of mapping anatomical surfaces with improved accuracy and greater flexibility. More broadly, the idea of using an adaptive parameter domain allows easy handling of a wide range of biomedical shapes.

Genetic architecture of floral traits in bee- and hummingbird-pollinated sister species of Aquilegia (columbine).

Interactions with animal pollinators have helped shape the stunning diversity of flower morphologies across the angiosperms. A common evolutionary consequence of these interactions is that some flowers have converged on suites of traits, or pollination syndromes, that attract and reward specific pollinator groups. Determining the genetic basis of these floral pollination syndromes can help us understand the processes that contributed to the diversification of the angiosperms. Here, we characterize the genetic architecture of a bee-to-hummingbird pollination shift in Aquilegia (columbine) using QTL mapping of 17 floral traits encompassing color, nectar composition, and organ morphology. In this system, we find that the genetic architectures underlying differences in floral color are quite complex, and we identify several likely candidate genes involved in anthocyanin and carotenoid floral pigmentation. Most morphological and nectar traits also have complex genetic underpinnings; however, one of the key floral morphological phenotypes, nectar spur curvature, is shaped by a single locus of large effect.

Control of connectivity and rigidity in prismatic assemblies.

How can we manipulate the topological connectivity of a three-dimensional prismatic assembly to control the number of internal degrees of freedom and the number of connected components in it? To answer this question in a deterministic setting, we use ideas from elementary number theory to provide a hierarchical deterministic protocol for the control of rigidity and connectivity. We then show that it is possible to also use a stochastic protocol to achieve the same results via a percolation transition. Together, these approaches provide scale-independent algorithms for the cutting or gluing of three-dimensional prismatic assemblies to control their overall connectivity and rigidity.

Shape analysis via inconsistent surface registration.

In this work, we develop a framework for shape analysis using inconsistent surface mapping. Traditional landmark-based geometric morphometr- ics methods suffer from the limited degrees of freedom, while most of the more advanced non-rigid surface mapping methods rely on a strong assumption of the global consistency of two surfaces. From a practical point of view, given two anatomical surfaces with prominent feature landmarks, it is more desirable to have a method that automatically detects the most relevant parts of the two surfaces and finds the optimal landmark-matching alignment between these parts, without assuming any global 1-1 correspondence between the two surfaces. Our method is capable of solving this problem using inconsistent surface registration based on quasi-conformal theory. It further enables us to quantify the dissimilarity of two shapes using quasi-conformal distortion and differences in mean and Gaussian curvatures, thereby providing a natural way for shape classification. Experiments on Platyrrhine molars demonstrate the effectiveness of our method and shed light on the interplay between function and shape in nature.

Area-Preserving Mapping of 3D Carotid Ultrasound Images Using Density-Equalizing Reference Map.

Atherosclerotic plaques are focal and tend to occur at arterial bends and bifurcations. To quantitatively monitor the local changes in the carotid vessel-wall-plus-plaque thickness (VWT) and compare the VWT distributions for different patients or for the same patients at different ultrasound scanning sessions, a mapping technique is required to adjust for the geometric variability of different carotid artery models. In this work, we propose a novel method called density-equalizing reference map (DERM) for mapping 3D carotid surfaces to a standardized 2D carotid template, with an emphasis on preserving the local geometry of the carotid surface by minimizing the local area distortion. The initial map was generated by a previously described arc-length scaling (ALS) mapping method, which projects a 3D carotid surface onto a 2D non-convex L-shaped domain. A smooth and area-preserving flattened map was subsequently constructed by deforming the ALS map using the proposed algorithm that combines the density-equalizing map and the reference map techniques. This combination allows, for the first time, one-to-one mapping from a 3D surface to a standardized non-convex planar domain in an area-preserving manner. Evaluations using 20 carotid surface models show that the proposed method reduced the area distortion of the flattening maps by over 80% as compared to the ALS mapping method. The proposed method is capable of improving the accuracy of area estimation for plaque regions without compromising inter-scan reproducibility.

Planar morphometrics using Teichmüller maps.

Inspired by the question of quantifying wing shape, we propose a computational approach for analysing planar shapes. We first establish a correspondence between the boundaries of two planar shapes with boundary landmarks using geometric functional data analysis and then compute a landmark-matching curvature-guided Teichmüller mapping with uniform quasi-conformal distortion in the bulk. This allows us to analyse the pair-wise difference between the planar shapes and construct a similarity matrix on which we deploy methods from network analysis to cluster shapes. We deploy our method to study a variety of Drosophila wings across species to highlight the phenotypic variation between them, and Lepidoptera wings over time to study the developmental progression of wings. Our approach of combining complex analysis, computation and statistics to quantify, compare and classify planar shapes may be usefully deployed in other biological and physical systems.

Conformal mapping of carotid vessel wall and plaque thickness measured from 3D ultrasound images.

Measurements of vessel-wall-plus-plaque thickness (VWT) from 3D carotid ultrasound have been shown to be sensitive to the effect of pharmaceutical interventions. Since the geometry of carotid arteries is highly subject-specific, quantitative comparison of the distributions of point-wise VWT measured for different patients or for the same patients at different ultrasound scanning sessions requires the development of a mapping strategy to adjust for the geometric variability of different carotid surface models. In this paper, we present an algorithm mapping each 3D carotid surface to a 2D carotid template with an emphasis on preserving the local geometry of the carotid surface by minimizing local angular distortion. The previously described arc-length scaling (AL) approach was applied to generate an initial 2D VWT map. Using results established in the quasi-conformal theory, a new map was computed to compensate for the angular distortion incurred in AL mapping. As the 2D carotid template lies on an L-shaped non-convex domain, one-to-one correspondence of the mapping operation was not guaranteed. To address this issue, an iterative Beltrami differential chopping and smoothing procedure was developed to enforce bijectivity. Evaluations performed in the 20 carotid surface models showed that the reduction in average angular distortion made by the proposed algorithm was highly significant (P = 2.06 × 10-5). This study is the first study showing that a bijective conformal map to a non-convex domain can be obtained using the iterative Beltrami differential chopping and smoothing procedure. The improved consistency exhibited in the 2D VWT map generated by the proposed algorithm will allow for unbiased quantitative comparisons of VWT as well as local geometric and hemodynamic quantities in population studies.